As mentioned in section REF  (and exemplified in figure REF ), the PFE method is subject to the same issues as PSE, where any stable upstream-travelling mode in the spectrum is considered as an unstable downstream-travelling mode, which may lead to a spurious growth of these modes. This becomes even more serious with the repetition of the spectrum in the PFE method: the spatial step must be chosen to damp the evanescent acoustic branch at \(Re(\alpha )=0\)  at low frequencies, and also its replica at \(Re(\alpha )=\alpha _s\)  at higher frequencies. Still, other modes may arise in the analysis that could impact the spatial march. In particular, the neutral upstream acoustic branch may lead to the inclusion of an upstream wave in the march, as shown in the previous section; since this wave is not evanescent, this does not lead to the growth of spurious waves, but may lead to non-physical results due to the parabolisation of the equations. In theory, the correct behaviour of the upstream waves (including their respective modulation) could only be obtained using the equations parabolised in the opposite direction, which would march the desired mode from a given position downstream until the nozzle. This issue with upstream waves becomes worse for frequencies at which the guided jet mode becomes stable [1]}, [2]}, [3]}, which usually occurs at frequencies higher than the screech frequency. In those cases, the spatial step must be severely increased to avoid considering this new mode in the march, which also affects the resulting modulated wavepacket.
